Track a Rubik's cube to implement a 3D joystick/mouse control. 1\. Get out an old Rubik's cube (or get one from your parents). 2\. Write a program to detect the center of each colored square. 3\. Group these centers into lines and then find the vanishing points for each face. 4\. Estimate the rotation angle and focal length from the vanishing points. 5\. Estimate the full 3D pose (including translation) by finding one or more \(3 \times 3\) grids and recovering the plane's full equation from this known homography using the technique developed by Zhang (2000). 6\. Alternatively, since you already know the rotation, simply estimate the unknown translation from the known 3D corner points on the cube and their measured 2D locations using either linear or non-linear least squares. 7\. Use the 3D rotation and position to control a VRML or 3D game viewer.

Understanding the concept of vanishing points is crucial when analyzing perspective in images. Vanishing points are specific points in an image where parallel lines, extending from the real world, appear to converge. For instance, think about standing on a straight road and looking at the horizon; the sides of the road seem to touch at a distant point. This is a vanishing point.When it comes to implementing a virtual 3D joystick, detecting vanishing points can be pivotal. For each face of a Rubik's cube, by finding the center of each colored square and grouping them into sets of parallel lines, you can identify where these lines converge in the image plane. These points, often located at infinity in the real world, are bound within the confines of the image due to the camera's perspective. Detecting and utilizing these points allow us to infer a lot of information about the spatial orientation of objects, like the angles of rotation relative to the camera. The accuracy of vanishing points detection impacts the subsequent steps of 3D pose estimation and homography recovery.

Delving into computational camera geometry introduces the mathematical study of the projections and mapping of the 3D world onto 2D images. The geometry can become extremely complex as it involves understanding how the 3D structure of the scene is transformed into a 2D photograph. Key elements in this transformation include the camera's focal length, sensor characteristics, and lens distortion.
Computational camera geometry is the foundation for algorithms that estimate 3D poses from 2D images such as those taken of a Rubik's cube. These algorithms rely on understanding the intrinsic (such as the focal length and optical center) and extrinsic (position and orientation of the camera in space) parameters of a camera. For the joystick control exercise, understanding computational camera geometry is essential for translating the rotations and movements of the Rubik's cube into accurate input signals for the 3D environment.

Moving on to 3D pose estimation, this involves deducing the position and orientation of an object in three-dimensional space from a two-dimensional image. The 'pose' encompasses both rotation (orientation) and translation (position) elements. For your Rubik's cube, this means figuring out precisely how it's turned and where it is relative to the camera. This step is a cornerstone for Augmented Reality (AR) experiences and robotic vision, where understanding an object's real-world coordinates and how it's placed in an environment is crucial for interaction.
To achieve this, one might first recover the homography of the cube's faces and then apply specialized techniques such as Zhang's method. The success of pose estimation is not just linked to the technical approach used but also to the quality and calibration of the camera setup as defined by computational camera geometry. Precise 3D pose estimation is particularly important when the Rubik's cube is used to navigate or interact in a virtual space, so translating its real-world maneuvers into digital signals must be done with high fidelity.

Finally, homography recovery is about mapping one view of an object (like a plane of the Rubik's cube) to another, preserving the straight lines of the object. To put it simply, a homography is a transformation that lets us relate two different perspectives of the same planar surface. For example, it relates the squares on the surface of the Rubik's cube in real-life to their distorted representation on the image captured by a camera.
Recovering the homography, as a part of the exercise, involves identifying a set of points on the cube that forms a grid and using this to find the full equation for the plane on which these points lie. This process involves complex mathematics that takes into account camera geometry. Using Zhang's method, a popular technique for this purpose, we can calibrate the camera and extract the required 3D information. The ability to recover accurate homography is integral to precise pose estimation, which in turn is critical for using the Rubik's cube as a 3D control device. An inaccurately recovered homography would lead to errors in the perceived movement and rotation of the cube, resulting in an imprecise or incorrect control in the virtual environment.